View metadata, citation and similar papers at core.ac.uk Topology Vol. 38, No. 1, pp. 141Ð165,brought 1999 to you by CORE ( 1998 Elsevier Science Ltd All rights reserved. Printedprovided in Great by BritainElsevier - Publisher Connector 0040-9383/99 $19.00#0.00 PII: S0040-9383(98)00007-X LAMINATIONS WITH TRANSVERSE STRUCTUREs A. CANDEL (Received 18 August 1996; in revised form 20 January 1998) 1. INTRODUCTION In this paper we study laminations by surfaces of three-dimensional manifolds and some of their properties. These are very interesting objects, particularly useful in understanding the topology of the manifold which has no incompressible surface. Gabai and Oertel have shown in [9] that many familiar properties of su¦ciently large three manifolds continue to hold in the more general setting of ‘laminated’ manifolds. A lamination of a three manifold has associated with it certain transverse structure, roughly the action of the fundamental group of the ambient manifold on the space of leaves of the lamination lifted to the universal cover. This space of leaves can be improved to a tree- like object [9, 17]. In a few words, one starts with a lamination of M, having nowhere dense support (and some other properties to be made precise later), and passes to the universal cover and a tree is constructed by taking one point for each gap or non-boundary leaf of the lamination. The action of the fundamental group of M permutes the leaves and the gaps of the lamination, so it induces an action on the tree. If the lamination has an invariant transverse measure, as is the case of laminations of surfaces, the tree has a metric which is left invariant by the action of the fundamental group of the ambient manifold. We may say that the lamination has a transverse isometric structure. There has been much work done concerning these type of actions. However, when considering laminations by surfaces in three manifolds, the existence of an invariant measure is a strong restriction both on the lamination and on the three manifold. For example, an essential lamination in the sense of [9] in a tiny (i.e. non-su¦ciently large) three manifold has no invariant transverse measure. Even if we are able to associate a tree supporting an action of the fundamental group of the manifold, the absence of invariant transverse measure means that the tree has no preferred metric and the action cannot be by isometries with respect to any metric we put on it. It seems then natural to try to put geometric structures on the tree which are preserved by the action. There are several types of structures one can put on a tree which are preserved by an action of a group; we briefly discuss them in the next two sections. One way of doing this is by thinking that isometric structures on a tree are related to the continuous group of translations of the real line. Therefore, we may consider the other two familiar continuous groups which act on the line, the a¦ne group and the group of homographic transformations. s Research partially supported by NSF Grant DMS-9100383. 141 142 A. Candel The a¦ne group naturally appears in laminations of three manifolds obtained by suspension of pseudo-Anosov homeomorphisms of surfaces, while the group of homo- graphies is found in manifolds which are circle bundles over surfaces. But in general, these types of actions are still very restrictive for applications to three manifolds. For instance, the existence of an a¦ne action implies that the manifold is su¦ciently large. Furthermore, either type of structure is rarely invariant by Dehn surgery. The correct type of structure generalizing the action by isometries comes naturally when considering the question of giving su¦cient conditions on a branched surface so that it carries a lamination (asked by Gabai and Oertel in [9] ). The case of existence of measured lamination has a complete answer in terms of the singular homology of the branched surface. These homological conditions can be generalized by considering similar ones in homology groups with local coe¦cients on the branched surface. Not only they guarantee that the branch surface carries a lamination, but also that the latter comes equipped with a transverse measure which is almost invariant in the sense that each transformation of the holonomy pseudogroup of the lamination multiplies it by a constant factor (in a sense to be made precise later). We call this type of measures projective transverse measures. All this is discussed in Sections 4Ð6, as well as other structural properties of laminations with projective measure. One of them is the resemblance of these laminations with minimal sets of smooth codimension one foliations. Other is that the multipliers of the transverse measure come from a representation of the fundamental group of an essential branched surface carrying the lamination. One cannot expect this representation to extend to one of the fundamental group of the ambient manifold. In any case, there is, associated to a lamination with a projective transverse measure, a dual cohomology class (in dimension one) of the ambient manifold with coe¦cients usually in a finite group. After this we discuss pure laminations, an improved version of essential laminations. Their fundamental property is that their associated tree like object is a real tree. Moreover, if the lamination has a projective transverse measure, then this tree has a piecewise linear structure which is preserved by the action of the fundamental group of the ambient manifold, and conversely. In the last section we study properties of the fundamental group of laminated three manifolds. They are exponential growth and existence of non commutative free subgroups. We conclude this introduction with a few words about related work. In trying to generalize actions by isometries one notices that transversely a¦ne foliations form the first step beyond transversally isometric ones. Moreover, a particular example of a lamination (occurring as a minimal set of a foliations) with the type of transverse measure considered here appears at the end of Dippolito’s paper [4] (which comprises previous examples of Sacksteder and of Hirsch). Related examples are in Inaba’s [14]. Independently, Oertel [20] has also developed the notion of a¦ne measures for laminations (here called projective). His paper has some interesting examples. 2. GEOMETRIC STRUCTURES ON TREES A real tree (or R-tree for short) is a metric space (¹, d) such that: f For any pair of points x, y in ¹ there is a segment with endpoints x and y. (A segment in the metric space (¹, d) is a subset of ¹ isometric to some interval in R.) f The intersection of two segments in ¹ with at least a point in common is a segment. f The union of two segments of ¹ whose intersection is a single point which is an endpoint of each is a segment. LAMINATIONS WITH TRANSVERSE STRUCTURE 143 Since we are not so much interested in the metric properties of a tree, it is convenient to recall the following characterization of trees proved in [15]. A (real) tree is a separable metrizable space which is locally arc wise connected and uniquely arc wise connected. In other words, a tree is a separable metrizable space with the property that for any pair of points in it there is a unique segment joining them. Given a tree with metric (¹, d) we have the group of isometries of ¹, that is homeomor- phisms of ¹ that preserve d, and we have the group of a¦ne transformations of ¹, that is homeomorphisms g of ¹ which preserve the metric up to a factor d(gx, gy)"j(g) d(x, y) for all x, y in ¹, j(g) some positive real number, called the stretch of g. Evidently, the notion of a¦ne action can be extended to "-trees. We denote by I(¹ ) the group of isometries and by A(¹ ) the group of a¦ne homeomor- phisms. Clearly I(¹ )LA(¹ ), and is a normal subgroup. Observe that the stretch factor map induces a homomorphism j from A(¹ ) to the multiplicative group of positive real numbers R . Furthermore, I(¹ )"Ker(j) , so that we * may think of A(¹ )/I(¹) as a subgroup of R . In particular, it is commutative. * An a¦ne homeomorphism g of ¹ with j(g)O1 has at most one fixed point on ¹, for if x and y are both fixed by g, then d(x, y)"d(gx, gy)"j(g) d(x, y) so that j(g)"1. We have an exact sequence of groups 1PI(¹ )PA(¹ )PA(¹ )/I(¹)P1 so that if g1, g2 are a¦ne transformations with the same stretch factor, they di¤er by an ~13 ¹ isometry: g1 ¡ g2 I( ). We briefly recall the classification of isometries of an R-tree. More details can be found, among other places, in [3]. f The isometry h has a fixed point. The fixed point set Fh of h is a subtree. f There is an axis for h. That is, there is an isometry from R into ¹ whose image is invariant by h and on which h acts as a translation by a positive amount q(h). The hyperbolic length (or simply length) l(h) of an element h of I(¹ ) is 0 in the first case and q(h) in the second. Let ! be a finitely presented group and ' : !PA(¹ ) a representation. The composition j ' ! ! ! log( ) ¡ defines an element of Hom ( , R). The commutator subgroup [ , ] has image contained in I(¹ ). As we show in the section of examples, it follows from Thurston’s theory of pseudo- Anosov homeomorphisms of surfaces that the fundamental group of a three-manifold that fibers over the circle with pseudo-Anosov monodromy acts a¦nely on a tree.